First slide

Geometry of complex numbers

Question

$z_{1}$ and $z_{2}$ are the roots of $z^{2} + a z + b = 0$ where $a, b$ are non-zero complex numbers. It is known that the line joining the points $A (z_{1})$ and $B (z_{2})$ pass through the origin, then $a^{2} / b$ is:

Moderate

A

Real
B

Purely imaginary
C

$\arg (a^{2} / b) = π / 4$
D

None of these

Solution

Line joining $z_{1}$ and $z_{2}$ passes through origin, therefore, $z_{2} = λ z_{1} f o r s o m e λ \in R$
We have $z_{1} + z_{2} = - a$ $\Rightarrow (1 + λ) z_{1} = - a$
and $z_{1} z_{2} = b$ $\Rightarrow λ z_{1}^{2} = b$
$∴$ $\frac{a^{2}}{b} = \frac{{(1 + λ)}^{2} z_{1}^{2}}{λ z_{1}^{2}} = \frac{{(1 + λ)}^{2}}{λ} = r e a l$

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